<span class='Hi'>Mark</span> Balaguer’s project in this book is extremely ambitious; he sets out to defend both platonism and ﬁctionalism about mathematical entities. Moreover, Balaguer argues that at the end of the day, platonism and ﬁctionalism are on an equal footing. Not content to leave the matter there, however, he advances the anti-metaphysical conclusion that there is no fact of the matter about the existence of mathematical objects.1 Despite the ambitious nature of this project, for the most part Balaguer does not shortchange the reader on rigor; all the main theses advanced are argued for at length and with remarkable clarity and cogency. There are, of course, gaps in the account (some of which are described below) but these should not be allowed to overshadow the sig-.